A Generalization on Some New Types of Hardy-Hilbert’s Integral Inequalities

Banyat Sroysang Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, PathumThani 12121, Thailand Correspondence should be addressed to Banyat Sroysang; banyat@mathstat.sci.tu.ac.th Received 15 May 2013; Accepted 17 September 2013 Academic Editor: Wilfredo Urbina Copyright © 2013 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Sulaiman presented, in 2008, new kinds of Hardy-Hilbert’s integral inequality in which the weight function is homogeneous. In this paper, we present a generalization on the kinds of Hardy-Hilbert’s integral inequality.

We also recall that a nonnegative function (, ) which is said to be homogeneous function of degree  if (, ) =   (, ) for all  > 0. And we say that (, V) is increasing if (1, ) and (, 1) are increasing functions.
Then, for all  > 0, one has where Proof.Let  > 0 and  = ∬  0 ()(V)/((, V)) V.By the Hölder inequality, the assumption of , and the Tonelli theorem, we have V) Now, we put  = V/ and  = V/ for the first integral, and then we put  = /V and  = /V for the second integral.
And, by Proposition 1, one has V) V) V) Then, by the assumption, one has This proof is completed.Then, for all  > 0, one has (c) This follows from Theorem 2 where () =  +  2 for all .

Applications
(d) This follows from Theorem 2 where () =   for all .

Open Problem
In this section, we pose a question that is how to generalize the integral inequality (13) if  may not satisfy the property () ≥  for all  > 0.